target="_blank" rel="nofollow" href="#ulink_bd26938e-6af3-5923-a68b-6d3226f8acee">Figure 3.2 (a) A precise value of the Zeeman energy difference between the two states in a spin-½ system should imply a single value in the transition, hence a delta function in the frequency distribution. (b) In reality, a wider line shape such as a Lorentzian or Gaussian suggests an uncertainty in the difference between the energy levels. For simple liquids, the line shape is a Lorentzian in the frequency domain, which corresponds to the exponential decay in the time-domain FID, shown in (c). A fast decay of the FID (e.g., short blue dash) implies a short T2 and a wide line shape; a slow decay (e.g., red solid line) implies a long T2 and a narrow line shape. A precise value of the energy difference as in (a) would imply a sinusoidal oscillation without any decay in the time domain (as shown in Figure 2.15d).
The Bloch equation [Eq. (2.18)] contains a phenomenological term leading to exponential relaxation. This classical description is quite accurate for spins in rapidly tumbling molecules but breaks down when the motions of molecules become slow or complex, such as in the case of internal motion in macromolecules. In the following sections, we first explain the relaxation mechanisms in terms of quantum transitions between eigenstates of operators Ix, Iy, and Iz; then, we briefly describe the results of the random field model of relaxation.
3.7.1 Relaxation Mechanism in Terms of Quantum Transitions
For spin-1/2 particles, the relaxation mechanism can be understood with a set of equations and analysis in terms of quantum transition [9, 10]. In this approach, the spin populations (the occupancies of the eigenstates of Iz with eigenvalues m=±1/2) are defined as
We also define the total population N0 and the population difference n as
Hence, the macroscopic magnetization M is proportional to the population difference n. Using Eq. (3.21), the z component of the magnetization at time = 0 can be written as
Since the population is at equilibrium with the environment according to the Boltzmann distribution, the population ratio is
To consider the dynamics of the two populations, we define w+- as the probability of transition of a spin from |+> state to |–> state per spin per second, and w-+ as the probability of transition of a spin from |–> state to |+> state per spin per second. At equilibrium, we have
Combining Eq. (3.28) and Eq. (3.29), we have
With this equation, the changes of the spins with time can be defined as
Each equation in Eq. (3.31) has two terms, the increment term (the first term) and the reduction term (the second term). Given the fact that w+- ≈ w-+, we define
Note that w0 can be considered as the probability of induced transitions, while the term ℏγB0/kBT can be considered as the probability of spontaneous transitions; their differences were distinguished first by Albert Einstein in 1916 when he published a paper on different processes occurring in the formation of an atomic spectral line in optical studies. And hence we can show that
where dn/dt goes to zero as n (a variable) goes to n∞ (a constant given by the population difference in the presence of B0 but in the absence of another rf field). If we define
and multiplying Eq.
